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  • Jun 24 22:14
    ChrisRackauckas commented #878
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Paweł Biernat
@pwl
that's great, we will let you know once we fix our docs, you could write some wrappers yourself if you are up for it:-)
Christopher Rackauckas
@ChrisRackauckas
So I'll get to there so you guys can take a look at it. If you think it's something that could be used for doing the rest of the wrapping (and we agree on whatever API changes), then we should merge into some org and get the rest of it together.
Just let me know if that "new" interface sol = ODE.solve(ode,stepper;opts...) ends up changing at all.
I'll be watching that dev branch.
Paweł Biernat
@pwl
@ChrisRackauckas you mentioned wave relaxation, do you have any good books/overview papers on that?
Christopher Rackauckas
@ChrisRackauckas
I know Burrage's book covers it.
I was talking with Kevin Burrage about it not too long ago. Let me find the reference.
I guess that tome is kind of expensive and out of print.
Paweł Biernat
@pwl
that is some obscure source
Christopher Rackauckas
@ChrisRackauckas
I'll probably pick up a copy myself, but I'll send Kevin an email and see if I can get a PDF, or what the "latest" reference is (I know this is from early 2000)
(Late 1990s)
Paweł Biernat
@pwl
cool, let me know if you get it
Christopher Rackauckas
@ChrisRackauckas
Well it's not obscure, you'd find it if you were reading literature and looked at citations.
But it's just out of print.
Paweł Biernat
@pwl
what I mean by it is that this doesn't seem like a popular/well known method if the only book about it is out of print
Christopher Rackauckas
@ChrisRackauckas
None of them really are.
I mean, you can find a bunch of articles on it.
Paweł Biernat
@pwl
have you tested it?
Christopher Rackauckas
@ChrisRackauckas
There are a bunch of articles which I think tend to show that Waveform Relaxation "kind of" came out on top of the other parallel methods (parallel extrapolation, special Runge-Kutta methods)
But it's supposedly a little difficult to program.
That's why I want to try it out.
I know Hairer has parallel extrapolation in ODEX
so there's no reason to code that
Paweł Biernat
@pwl
thanks!
Christopher Rackauckas
@ChrisRackauckas
The special Runge-Kutta methods you just need to use threading. They just turn it into solving Ax=b each step in some simplified way.
Christopher Rackauckas
@ChrisRackauckas
@pwl This is a good SE post for explaining the difference between the major groups., and here is another reference for the differences . We might think about writing a wrapper to PFASST. Actually I think it we wrap enough of them or put enough of these algorithms under the same roof we might be able to get a review article out of it.
In the end I am looking to try a few of them since I am interesting in which ones may be useful for stochastic differential equations.
But I don't really think that any program puts all of these together already, so this is something to be done.
While I'm at it, the other ODE methods I am looking into implementing are super high order RKs.
Let me know if you plan on implementing any of these so we don't double up!
Paweł Biernat
@pwl
have you looked at high order Taylor methods?
Christopher Rackauckas
@ChrisRackauckas
I mean, I know how they're derived.
Paweł Biernat
@pwl
There was some package on github (now removed), that implemented arbitrary order Taylor methods, which I hooked up to ODE to test the backend API and the method, they worked!
Christopher Rackauckas
@ChrisRackauckas
Did you have to provide the derivatives?
Paweł Biernat
@pwl
ForwardDiff does it for you
Christopher Rackauckas
@ChrisRackauckas
But for really high derivatives, that doesn't give more overhead?
Paweł Biernat
@pwl
it does, but on the other hand you can make longer stepsizes
Christopher Rackauckas
@ChrisRackauckas
Depending on the function being "nice"
Paweł Biernat
@pwl
so it sort of balances out
Christopher Rackauckas
@ChrisRackauckas
With automatic differentiation... that's interesting. I'd like to test that out vs other methods.
Paweł Biernat
@pwl
I tested it with a hyperbolic PDE that I work on and it worked as quick as RK
Christopher Rackauckas
@ChrisRackauckas
I thought of using symbolic differentiation before with Taylor methods.
Paweł Biernat
@pwl
at least the same order of magnitude
the fun thing with this method is that you can prescribe an error much lower then the machine epsilon
like reltol 1e-30 with Float64
Christopher Rackauckas
@ChrisRackauckas
Probably scales better than extrapolation.
Paweł Biernat
@pwl
but this is a niche application
Christopher Rackauckas
@ChrisRackauckas
Well you'd need to go to other numbers to avoid weird truncation error problems at that point.